Dynamics in the space of metrics and new invariants in ergodic theory

1. Admissible metrics on the measure space: new trend in the theory of mm spaces.
2. Classification of mm-spaces. Matrix distributions.
3. Thе actions of measure preserving groups in the space of admissible metrics in measure space. ($\mathcal{M} \subset L^1(X\otimes X, \mu \otimes \mu)$).
4. Average metrics, ergodic limit and asymptotic invariants,
5. Sclaing entropy. Scaling entropy function. Examples.
6. Theorem. Bounded scaling entropy and discrete spectra. Sequential entropy by Kushnirenko. All possible scalings.
7. New geometrical problems.